Apparent contours of nonsingular real cubic surfaces

We give a complete deformation classification of real Zariski sextics, that is of generic apparent contours of nonsingular real cubic surfaces. As a by-product, we observe a certain "reversion" duality in the set of deformation classes of these sextics.Comment: 61 pages, 8 figures, Revised version to be published in Transactions AMS: some minor corrections, a missing lemma is include

Enumerative Real Algebraic Geometry

Enumerative Geometry is concerned with the number of solutions to a structured system of polynomial equations, when the structure comes from geometry. Enumerative real algebraic geometry studies real solutions to such systems, particularly a priori information on their number. Recent results in this area have, often as not, uncovered new and unexpected phenomena, and it is far from clear what to expect in general. Nevertheless, some themes are emerging. This comprehensive article describe the current state of knowledge, indicating these themes, and suggests lines of future research. In particular, it compares the state of knowledge in Enumerative Real Algebraic Geometry with what is known about real solutions to systems of sparse polynomials.Comment: Revised, corrected version. 40 pages, 18 color .eps figures. Expanded web-based version at http://www.math.umass.edu/~sottile/pages/ERAG/index.htm

Arthur Cayley as Sadleirian Professor: A Glimpse of Mathematics Teaching at 19th-Century Cambridge

AbstractThis article contains the hitherto unpublished text of Arthur Cayley's inaugural professorial lecture given at Cambridge University on 3 November 1863. Cayley chose a historical treatment to explain the prevalent basic notions of analytical geometry, concentrating his attention in the period (1638–1750). Topics Cayley discussed include the geometric interpretation of complex numbers, the theory of pole and polar, points and lines at infinity, plane curves, the projective definition of distance, and Pascal's and Maclaurin's geometrical theorems. The paper provides a commentary on this lecture with reference to Cayley's work in geometry. The ambience of Cambridge mathematics as it existed after 1863 is briefly discussed.Copyright 1999 Academic Press.Cet article contient le texte jusqu'ici inédit de la leçon inaugurale de Arthur Cayley donnée à l'Université de Cambridge le 3 novembre 1863. Cayley choisit une approche historique pour expliquer les notions fondamentales de la géométrie analytique, qui existaient alors, en concentrant son attention sur la période 1638–1750. Les sujets discutés incluent l'interpretation géométrique des nombres complexes, la théorie des pôles et des polaires, les points et les lignes à l'infini, les courbes planes, la définition projective de la distance, et les théorèmes géométriques de Pascal et de Maclaurin. L'article contient aussi un commentaire reliant cette leçon à l'oeuvre de Cayley en géométrie. L'atmosphère des mathématiques à Cambridge après 1863 est brièvement discutée.Copyright 1999 Academic Press.MSC Classification: 01A55, 01A72, 01A73

Quilting Topological Phases of Matter with Quantum Thread: A Luttinger Liquid Love Letter

Kicked off by the discovery of the quantum Hall effect in the early 1980s, the study of topological phases of matter has captured the attention of the condensed matter physics community for over four decades. With topologically ordered phases, symmetry-protected topological phases and, most recently, fracton phases, examples of states of matter beyond the Landau-Ginzburg symmetry breaking paradigm abound. One approach for constructing these novel states of matter is to employ a layered approach; 2-dimensional phases can be built by coupling together 1-dimensional wires , 3-dimensional phases can be built by coupling together 2-dimensional layers and/or 1-dimensional wires and so on. Two major advantages of this approach are its analytical tractability and its ability to describe chiral phases. In this dissertation we will make use of these constructions to study several new and exotic strongly coupled quantum phases of matter. These include necessarily interacting fermionic symmetry-protected topological phases, chiral fracton phases and stable compressible phases which lack any local order parameter

Revolutions in Parallel: The Rise and Fall of Drawing in Architectural Design.

This dissertation examines how the foundational principles of architectural design are influenced and reflected the discipline’s conceptual media. The first section explores the transition to drawing as architecture’s conceptual medium. Arguing that the use of drawing within masonic traditions of the Gothic period was not the same as its use during the early Renaissance, this work maintains that the simultaneous employment of plan, section and elevation (i.e. triadic form) was key to changing how drawing was understood and utilized in design. Examinations of Strasbourg Plan A (c. 1260) and the Milan Cathedral Plan and Section (c. 1390) demonstrate how drawings that appear orthographic may not indicate the use of orthography to prefigure forms in space. The examination of Raphael’s interior drawing of the Pantheon (c. 1509) further demonstrates that more than just a technical hurdle, the use of triadic form indicates epistemic shifts in both the understanding of design as a human rather than exclusively divine activity, and in the elevation of form as the primary quality of architectural contemplation. The second section of this dissertation examines the transition to computation as the medium of design. Through an exploration of Peter Eisenman’s House VI (c. 1975), this section demonstrates that the shift towards process-based (as opposed to form-based) thinking isn’t dependant on computation as a medium, and yet the medium of drawing constrains the ways in which process can contemplated. Further, this section suggests that rather than being a twentieth-century development, a turn to process is evidenced during the nineteenth-century by emerging fields like morphology, biology and genetics. Gehry Technologies’ project for the Yas Island Formula-One Hotel and Evan Douglis’ project for Choice Restaurant (both 2009) demonstrate how the focus on process and the use of computation as a medium impact both the practice and aesthetics of architecture. Tying these sections together, the over-arching argument of this work is that these two shifts in medium are similar in scope and impact for the architectural discipline. Like the transition to drawing centuries before, today’s shift to computation imbricates both technical and epistemological developments for the representation, design and practice of architecture.Ph.D.ArchitectureUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/64659/1/kluce_1.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/64659/2/kluce_2.pdfhttp://deepblue.lib.umich.edu/bitstream/2027.42/64659/3/kluce_3.pd